Here's how it went for me, in case anyone is interested. There were no additional emoji to be had on my Edge and my Brave browsers (just the thumbs up), so I tried Chrome. The thing is, while they were all there on Chrome, they were so small that I could hardly make them out, so I finally downloaded and switched to Firefox for CChat and that solved the problem.
This was several weeks ago, but the CChat emoji problems continue on Edge, Brave and Chrome, just FYI.